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<!-- ==================== CLASS DESCRIPTION ==================== -->
<h1 class="epydoc">Class Hessian</h1><p class="nomargin-top"><span class="codelink"><a href="numdifftools.core-pysrc.html#Hessian">source&nbsp;code</a></span></p>
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<pre class="literalblock">
Estimate Hessian matrix, with error estimate

Parameters
----------
fun : callable
    function to differentiate.
n : Integer from 1 to 4 defining derivative order.     (Default 1)
order : Integer from 1 to 4 defining order of basic method used.
        For 'central' methods, it must be from the set [2,4]. (Default 2)
method : Method of estimation.  Valid options are:
        'central', 'forward' or 'backward'.          (Default 'central')
romberg_terms : Number of Romberg terms used in the extrapolation.
        Must be an integer from 0 to 3.                       (Default 2)
        Note: 0 disables the Romberg step completely.
step_fix : If not None, it will define the maximum excursion from step_nom
             that is used and prevent the adaptive logic from working.
             This will be considerably faster, but not necessarily
             as accurate as allowing the adaptive logic to run.
            (Default: None)
step_max  : Maximum allowed excursion from step_nom as a multiple of it. (Default 4)
step_nom  : Nominal step.                          default maximum(x0, 0.02) 
step_ratio: Ratio used between sequential steps in the estimation
             of the derivative (Default 2)
        The steps used h_i = step_nom[i]*step_max*step_ratio**(-arange(steps_num))
step_num : integer
    if not specified it will be set according to the following rules: 
        step_num = 26 if step_fix is None
        step_num = 3.+ np.ceil(self.n/2.) + self.order + self.romberg_terms +4 otherwise
vectorized : True  - if your function is vectorized.
            False - loop over the successive function calls (default).

Uses a semi-adaptive scheme to provide the best estimate of the
derivative by its automatic choice of a differencing interval. It uses
finite difference approximations of various orders, coupled with a
generalized (multiple term) Romberg extrapolation. This also yields the
error estimate provided. See the document DERIVEST.pdf for more explanation
of the algorithms behind the parameters.

 Note on order: higher order methods will generally be more accurate,
         but may also suffer more from numerical problems. First order
         methods would usually not be recommended.
 Note on method: Central difference methods are usually the most accurate,
        but sometimes one can only allow evaluation in forward or backward
        direction.



HESSIAN estimate the matrix of 2nd order partial derivatives of a real
valued function FUN evaluated at X0. HESSIAN is NOT a tool for frequent
use on an expensive to evaluate objective function, especially in a large
number of dimensions. Its computation will use roughly  O(6*n^2) function
evaluations for n parameters.

Assumptions
-----------
fun : SCALAR analytical function
    to differentiate. fun must be a function of the vector or array x0,
    but it needs not to be vectorized.

x0 : vector location
    at which to differentiate fun
    If x0 is an N x M array, then fun is assumed to be a function
    of N*M variables.

Examples
--------
&gt;&gt;&gt; import numpy as np
&gt;&gt;&gt; import numdifftools as nd

# Rosenbrock function, minimized at [1,1]
&gt;&gt;&gt; rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
&gt;&gt;&gt; Hfun = nd.Hessian(rosen)
&gt;&gt;&gt; h = Hfun([1, 1]) 
&gt;&gt;&gt; h
array([[ 842., -420.],
       [-420.,  210.]])
&gt;&gt;&gt; Hfun.error_estimate &lt; 1.e-11
array([[ True,  True],
       [ True,  True]], dtype=bool)

# cos(x-y), at (0,0)
&gt;&gt;&gt; cos = np.cos
&gt;&gt;&gt; fun = lambda xy : cos(xy[0]-xy[1])
&gt;&gt;&gt; Hfun2 = nd.Hessian(fun)
&gt;&gt;&gt; h2 = Hfun2([0, 0]) 
&gt;&gt;&gt; h2
array([[-1.,  1.],
       [ 1., -1.]])
&gt;&gt;&gt; np.abs(h2-np.array([[-1,  1],[ 1, -1]])) &lt; Hfun2.error_estimate
array([[ True,  True],
       [ True,  True]], dtype=bool)

&gt;&gt;&gt; Hfun2.romberg_terms = 3
&gt;&gt;&gt; h3 = Hfun2([0,0])
&gt;&gt;&gt; h3
array([[-1.,  1.],
       [ 1., -1.]])
&gt;&gt;&gt; np.abs(h3-np.array([[-1,  1],[ 1, -1]])) &lt; Hfun2.error_estimate
array([[ True,  True],
       [ True,  True]], dtype=bool)


See also
--------
Gradient,
Derivative,
Hessdiag,
Jacobian

</pre>

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          <td><span class="summary-sig"><a href="numdifftools.core.Hessian-class.html#hessian" class="summary-sig-name">hessian</a>(<span class="summary-sig-arg">self</span>,
        <span class="summary-sig-arg">x00</span>)</span><br />
      Hessian matrix i.e., array of 2nd order partial derivatives</td>
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            <span class="codelink"><a href="numdifftools.core-pysrc.html#Hessian.hessian">source&nbsp;code</a></span>
            
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    <p class="indent-wrapped-lines"><b>Inherited from <code><a href="numdifftools.core.Hessdiag-class.html">Hessdiag</a></code></b>:
      <code><a href="numdifftools.core.Hessdiag-class.html#hessdiag">hessdiag</a></code>
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        <a href="numdifftools.core.Hessian-class.html#_hessian_txt" class="summary-name" onclick="show_private();">_hessian_txt</a> = <code title="'''
    Parameters
    ----------
    fun : callable
        function to differentiate.
    n : Integer from 1 to 4 defining derivative order.     (Default 1)
    order : Integer from 1 to 4 defining order of basic method used.
            For \'central\' methods, it must be from the set [2,4]. (D\
..."><code class="variable-quote">'</code><code class="variable-string">\n    Parameters\n    ----------\n    fun : ca</code><code class="variable-ellipsis">...</code></code>
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    </td><td class="summary">
        <a href="numdifftools.core.Hessian-class.html#__doc__" class="summary-name">__doc__</a> = <code title="''' Estimate Hessian matrix, with error estimate
    '''+ _hessian_txt+ '''

    HESSIAN estimate the matrix of 2nd order partial derivatives of a \
real
    valued function FUN evaluated at X0. HESSIAN is NOT a tool for fre\
quent
    use on an expensive to evaluate objective function, especially in \
...">''' Estimate Hessian matrix, with error estima<code class="variable-ellipsis">...</code></code>
    </td>
  </tr>
  <tr>
    <td colspan="2" class="summary">
    <div class="private">    <p class="indent-wrapped-lines"><b>Inherited from <code><a href="numdifftools.core.Hessdiag-class.html">Hessdiag</a></code></b> (private):
      <code><a href="numdifftools.core.Hessdiag-class.html#_hessdiag_txt" onclick="show_private();">_hessdiag_txt</a></code>
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  <h3 class="epydoc"><span class="sig"><span class="sig-name">__call__</span>(<span class="sig-arg">self</span>,
        <span class="sig-arg">x00</span>)</span>
    <br /><em class="fname">(Call operator)</em>
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<a name="hessian"></a>
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  <h3 class="epydoc"><span class="sig"><span class="sig-name">hessian</span>(<span class="sig-arg">self</span>,
        <span class="sig-arg">x00</span>)</span>
  </h3>
  </td><td align="right" valign="top"
    ><span class="codelink"><a href="numdifftools.core-pysrc.html#Hessian.hessian">source&nbsp;code</a></span>&nbsp;
    </td>
  </tr></table>
  
  <p>Hessian matrix i.e., array of 2nd order partial derivatives</p>
  <p>See also derivative, gradient, hessdiag, jacobian</p>
  <dl class="fields">
  </dl>
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  <h3 class="epydoc">_hessian_txt</h3>
  
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    <dt>Value:</dt>
      <dd><table><tr><td><pre class="variable">
<code class="variable-quote">'''</code><code class="variable-string"></code>
<code class="variable-string">    Parameters</code>
<code class="variable-string">    ----------</code>
<code class="variable-string">    fun : callable</code>
<code class="variable-string">        function to differentiate.</code>
<code class="variable-string">    n : Integer from 1 to 4 defining derivative order.     (Default 1)</code>
<code class="variable-string">    order : Integer from 1 to 4 defining order of basic method used.</code>
<code class="variable-string">            For \'central\' methods, it must be from the set [2,4]. (D</code><span class="variable-linewrap"><img src="crarr.png" alt="\" /></span>
<code class="variable-ellipsis">...</code>
</pre></td></tr></table>
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  </dl>
  <dl class="fields">
    <dt>Value:</dt>
      <dd><table><tr><td><pre class="variable">
''' Estimate Hessian matrix, with error estimate
    '''+ _hessian_txt+ '''

    HESSIAN estimate the matrix of 2nd order partial derivatives of a <span class="variable-linewrap"><img src="crarr.png" alt="\" /></span>
real
    valued function FUN evaluated at X0. HESSIAN is NOT a tool for fre<span class="variable-linewrap"><img src="crarr.png" alt="\" /></span>
quent
    use on an expensive to evaluate objective function, especially in <span class="variable-linewrap"><img src="crarr.png" alt="\" /></span>
<code class="variable-ellipsis">...</code>
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